1. Field of the Invention
The present invention relates to methods and apparatus for surface and wavefront topology; more particularly, to methods and apparatus for mathematically stitching together a plurality of overlapping individual sub-aperture maps to yield a full-aperture map of a surface; and most particularly, to such methods and apparatus wherein accuracy and resolution are increased through constrained simultaneous stitching of all sub-aperture data sets.
2. Discussion of the Related Art
The basic goal of sub-aperture metrology is to obtain a full-aperture measurement of a test part, without having to measure the entire part at one time. Because the relative position of each sub-aperture measurement is not known exactly, there is some ambiguity when combining the individual sub-apertures into a full aperture map. Accurately resolving this ambiguity is the fundamental task faced by all sub-aperture stitching methods. The uncertainty is largely due to alignment errors (small, unknown displacements) and noise in the individual sub-aperture maps. Sub-aperture stitching is thus an optimization problem: the goal is the minimization of the discrepancy between multiple data sets by including components related to various alignment errors. These added components are referred to herein as “compensators” and their form is evidently fixed by considering small displacements of, say, the test surface. For example, tilting a flat surface adds a linear component to the surface measurement (along the direction of the tilt). Some important features of a particular stitching method are the figure of merit and the compensators that it employs.
Sub-aperture interferometry was originally introduced to avoid the expense of fabricating large reference optics for testing astronomical mirrors. Publications in this area through the 1980's combine non-overlapping (sub-aperture) data sets into a smooth global map (i.e. they assume a priori knowledge that the test surface is smooth). In effect, the figure of merit is chosen to reflect the notion of “global smoothness”. Three or four compensators are used: piston and two tilts, with power also included when testing non-flat surfaces. These components are chosen for each sub-aperture so that, when tiled appropriately, the modified data sets appear to give windowed views into a map that is globally smooth. The high-resolution information of the individual maps is typically lost, as only a global polynomial fit is retained.
More recently, it is known to overlap the individual sub-aperture data sets. The unknown additive piston and tilts in each sub-aperture can then be determined by maximizing the data consistency in the overlap regions (i.e. still only 3 compensators, but the figure of merit becomes self evident). What at first appears to be redundant data in the overlap regions (e.g. multiple data values at many points on the test surface) is actually the key to stitching sub-aperture data sets together more effectively. This criterion of self-consistency replaces the more nebulous a priori requirement for “global smoothness”. As a result, rather than extracting just a global polynomial fit to the data, the high-resolution information of the individual maps can be retained.
It is known to use an automated stitching interferometer for testing large (>400 mm) flat optics. Also known is the importance of first performing a high-precision calibration of the reference flats. Significant new challenges appear, however, when these sub-aperture stitching techniques are applied to non-plane surfaces.
For example, sub-aperture measurement techniques have also been applied to rotationally-symmetric aspheres. It has been proposed to take a series of interferograms at different longitudinal positions of the test surface. Each interferogram gives an annular sub-aperture of useful data (i.e. within a zone centered on the asphere's axis). In this case, power compensation must be associated with each of the zonal data sets (in addition to piston and tilts) to create a consistent global result. This extra freedom follows from the fact that the test wavefronts are now spheres of different curvature, and relative motion thus generates a new effect. Because of this extra freedom, the process is not as well conditioned: that is, there is a greater sensitivity to errors and noise. The accuracy of the end results consequently suffer, especially in the rotationally symmetric components of the surface figure.
U.S. Pat. No. 5,416,586 discloses methods for annular stitching of aspheres. These methods include overlap between the different annuli to reduce the error sensitivity. These annular stitching methods extend the testing capability only to larger aspheric departures; they do not address the particular challenges associated with the testing of large-aperture surfaces, nor are they particularly robust in the presence of noise and misalignments. This is especially so if the amount of overlap is reduced to moderate levels.
U.S. Pat. No. 5.991,461 discloses a related technique for testing flats with an interference microscope. The basic idea remains the same, only on a reduced scale: individual sub-apertures now can have dimensions of the order of one millimeter or less. The unknown pistons and tilts in each sub-aperture are compensated, while retaining the high-resolution data. This method uses progressive pair-wise stitching (the tilt and piston are determined one sub-aperture at a time). Such a process is sensitive to the order of operations, and stitching errors can accumulate. In this particular method, the number of “good” data points in each overlap region determines the order of stitching. The overlap region with the most valid data is stitched first. Successive sub-apertures are chosen that have the most valid points in common with the data that have already been stitched. Other than suggesting a certain level of overlap, this method does not include any methodology for error reduction. In particular, sub-aperture placement errors are implicitly assumed to be negligible. Higher resolution measurements will either be contaminated by such errors, or require correspondingly accurate mechanical stages.
A related known method for stitching microscope data also utilizes progressive pair-wise stitching, but with a more sophisticated algorithm. A quantitative statistical criterion determines the order of stitching. Then the data most likely to be stitched well (under this criterion) are stitched first, and this helps to reduce error accumulation. Further, it is also known to use correcting sub-aperture placement errors, using unknown lateral and rotational components as well as the “conventional” linear compensators (piston and tilts in this case) which turns the stitching process into a computationally demanding non-linear problem. This method thus cannot stitch all data sets simultaneously (which is less sensitive to error accumulation than pair-wise stitching). While conventional “noise” terms, such as video noise, PZT miscalibration, vibration, etc., can affect the quality of the result, they can be suppressed somewhat in the stitching. However, even with “perfect” stitching, systematic errors (such as a non-flat reference surface, imaging aberrations and distortion) can accumulate and be magnified by this traditional stitching process. Addressing such matters is an important component of the present invention.
U.S. Pat. No. 5,960,379 discusses placement compensation in stitching. A six-dimensional position compensation scheme is proposed (three translations and three rotations). The authors then describe how using all six give unreasonable results. They find that all but one of the positional compensators are highly sensitive to noise in the relatively small overlap regions. Because optimization over all six compensating variables is so ill conditioned, they recommend using only the most significant of the six (i.e. deviation along the surface normal). This improves upon the accuracy achieved with six compensators, but the overall accuracy of the stitching technique then falls well short of that achieved on an individual sub-aperture measurement. In contrast, the present invention employs many compensators, with a constrained optimization, to ensure that the process is robust and the results are accurate.
Regarding calibration, the simplest calibration technique is the use of a reference standard. If a “perfect” surface is tested on some metrology instrument, only the measurement bias and noise of the instrument are evident in the measurement. Essentially the test and reference surfaces exchange roles for this calibration measurement, and averaging over multiple measurements can help to reduce any random components. Subsequent measurements can then approach the accuracy of the “perfect” surface by subtracting the calibration measurement. Such methods naturally require a high-quality and stable reference standard of sufficient size. Furthermore, thermal effects and changes in the instrument alignment can invalidate the calibration. Because of these difficulties, a variety of alternative prior art approaches to calibration have been developed and refined for interferometers.
The earliest such technique is known in the art as the “three-flat” test. Three surfaces are required, and they are first tested against each other in pairs (3 measurements). One of the surfaces is then rotated by 180 degrees and re-tested. Combining these four measurements allows for “absolute” data along one-dimensional stripes on the surfaces. The basic method has been advanced by using a Zernike surface representation to give two-dimensional results, and the method has been further extended to test spheres in addition to flats. The data processing of these methods, however, typically involves polynomial fitting (thereby reducing the resolution in the end results).
Another prior art technique applies only to the testing of spherical surfaces of non-zero curvature. Unlike the three-flat test, only two surfaces are required, ie. just the reference and test surfaces. The test part is measured in three different configurations: (i) the original (confocal) position, (ii) again confocal but rotated by 180 degrees about the optical axis, and (iii) the so-called cat's eye position (for which the test surface sits at the focus of the reference wave). The interferometer's system error is removed by combining the measurement data in a prescribed fashion. Calibration can be improved by taking additional measurements at the confocal position: by using four measurements with 90 degree offsets, it is possible to alleviate a misalignment contribution. Of course, this method does not apply to the calibration of either transmission flats or divergers, because neither has an accessible cat's eye position.
A further common technique of calibrating interferometers takes advantage of statistics. A calibration part is used, but it need not be “perfect” (unlike the reference standards discussed above). Instead, it should have a random distribution of error past some statistical correlation length. This means that two measurements, displaced spatially by the correlation length (or greater), are statistically uncorrelated. In applying this method to interference microscopy, a relatively high quality flat is measured in a variety of positions (separated by the correlation length of the surface or greater) on the interference microscope. All N measurements are then averaged together, providing an approximation of the interferometer system error. This approximation is (theoretically) good to the quality of the “reference” surface, divided by √{square root over (N)}. This general averaging technique has also been applied to systems with larger apertures, though lower spatial frequency features (“figure/form errors”) tend to be correlated. Nevertheless, this method has been used to cross-check other calibration methods for LIGO optics. Additionally, this technique has been employed on ball bearings to calibrate spherical reference surfaces.
U.S. Pat. No. 5,982,490 discloses an interesting variant that combines aspects of the aforementioned methods. The averaging technique is employed to obtain the asymmetric component of the surface. To get the rotationally symmetric component, however, a translated measurement is needed. The figure error is then computed with the aid of a polynomial fit. Translating the optic introduces some complications, though, particularly with distortion and edge effects (which the patent addresses to some extent). Because it is extracted as a polynomial fit, only a low-resolution map of the rotationally symmetric component of figure error is determined in this way. Although the approach disclosed in the present invention also involves rotating and translating the test surface, it is done in a manner that allows polynomial fitting to be applied only to the reference surface. This means that in the present invention the final results for the test surface can effectively retain the full resolution of the sub-aperture maps. This is especially effective when the mid- and high-frequency figure errors on the interferometer optics and the reference surface are much weaker than those on the test part.
In summary of the limitations of the prior art, despite the volume of prior art in stitching interferometry, no general stitching solution is currently available for non-flat surfaces. Large flats can be stitched without loss of resolution using a fairly robust algorithm (minimize mean-square error in all overlap regions simultaneously). The stage positioning accuracy required to keep the error magnification within reasonable bounds can be achieved readily since only two axes are needed; correction of positioning errors in software is thus unnecessary. Interference microscope data has also been stitched for smaller flats. Again, the lateral range is extended without loss of resolution, but there are accuracy losses. Pair-wise stitching algorithms, in particular, allow errors to accumulate. Stitching has also been performed on aspheres, but only to extend the aspheric departure that can be measured; this “zonal” stitching provides no gain in lateral resolution or range, and again comes at the price of accuracy losses. Although the prior art includes the possibility of position error correction within stitching, no one has reported an effective way to achieve this when stitching all sub-apertures simultaneously. The prior art in calibration also offers dedicated solutions to the problems associated with some global system errors (such as reference wave error). However, these problems can be addressed more effectively with a platform and method designed for stitching interferometry. Further, other components that are vital for accurate stitching (such as distortion correction) have not been addressed. A number of innovations are therefore needed to create an effective multi-axis system for accurate sub-aperture tests of non-flat parts.
It is a primary objective of the invention to synthesize a full-aperture numerical data map of a test surface of an object from a plurality of overlapping sub-aperture data maps of the surface.